NIMCET 2007 — Mathematics PYQ

To solve this efficiently, we can use a clever property of the characteristic equation rather than calculating individual eigenvalues.

Step 1: Understand the Characteristic Equation

The characteristic equation of a matrix $A$ is given by:

$$|A-\lambda I|=0$$

The roots of this cubic equation are the eigenvalues ${\lambda }_1,{\lambda }_2,$ and ${\lambda }_3$. Therefore, the characteristic polynomial can be written as:

$$|A-\lambda I|=({\lambda }_1-\lambda )({\lambda }_2-\lambda )({\lambda }_3-\lambda )$$

Step 2: Match the Target Expression

We need to find the value of:

$$E=(1+{\lambda }_1)(1+{\lambda }_2)(1+{\lambda }_3)$$

If we substitute $\lambda =-1$ into the characteristic polynomial identity:

$$|A-(-1)I|=({\lambda }_1-(-1))({\lambda }_2-(-1))({\lambda }_3-(-1))$$

$$|A+I|=(1+{\lambda }_1)(1+{\lambda }_2)(1+{\lambda }_3)$$

So, the value we are looking for is simply the determinant of the matrix $(A+I)$.

Step 3: Construct Matrix $(A+I)$

Add $1$ to the diagonal elements of matrix $A$:

$$A+I=\left[\begin{array}{ccc}1+1& amp;2& amp;15\\ 3& amp;4+1& amp;11\\ 5& amp;6& amp;7+1\end{array}\right]=\left[\begin{array}{ccc}2& amp;2& amp;15\\ 3& amp;5& amp;11\\ 5& amp;6& amp;8\end{array}\right]$$

Step 4: Calculate the Determinant $|A+I|$

Now, let's expand the determinant along the first row:

$$|A+I|=2\left|\begin{array}{cc}5& amp;11\\ 6& amp;8\end{array}\right|-2\left|\begin{array}{cc}3& amp;11\\ 5& amp;8\end{array}\right|+15\left|\begin{array}{cc}3& amp;5\\ 5& amp;6\end{array}\right|$$

Let's compute each $2\times 2$ determinant:

1. $\left|\begin{array}{cc}5& amp;11\\ 6& amp;8\end{array}\right|=(5\times 8)-(11\times 6)=40-66=-26$

2. $\left|\begin{array}{cc}3& amp;11\\ 5& amp;8\end{array}\right|=(3\times 8)-(11\times 5)=24-55=-31$

3. $\left|\begin{array}{cc}3& amp;5\\ 5& amp;6\end{array}\right|=(3\times 6)-(5\times 5)=18-25=-7$

Substitute these values back into the expansion:

$$|A+I|=2(-26)-2(-31)+15(-7)$$

$$|A+I|=-52+62-105$$

$$|A+I|=10-105$$

$$|A+I|=-95$$

Conclusion

$$(1+{\lambda }_1)(1+{\lambda }_2)(1+{\lambda }_3)=-95$$

Correct Option: (b) $-95$